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In this work, we will investigate the quadratic optimal control for bilinear systems. We will first study the existence of a solution for the considered optimal control. Then, we will focus on a special class of bilinear systems for which the quadratic optimal control can be expressed in a feedback law form. The approach relies on the conditions of optimality and linear semi-group theory.

In this paper, we establish the sharp L p boundedness for the bilinear integral operators with certain hypersingularities that generalize the bilinear Hilbert transform.

A Shallow Water Wave-like nonlinear differential equation is considered by using the generalized bilinear equation with the generalized bilinear derivatives D3,x and D3,t, which possesses the same bilinear form as the standard shallow water wave bilinear equation. By symbolic computation, four presented classes of rational solutions contain all rational solutions to the resulting Shallow Water Wave-like equation, which generated from a search for polynomial solutions to the corresponding generalized bilinear equation.

Commutators of bilinear pseudodifferential operators and the operation of multiplication by a Lipschitz function are studied. The bilinear symbols of the pseudodifferential operators considered belong to classes that are shown to properly contain certain bilinear Hormander classes of symbols of order one. The corresponding commutators are proved to be bilinear Calderon–Zygmund operators.

The optimization of a bilinear functional related to a linear state system with a modular control constraint is considered. Exact formulas for the functional increment are used to obtain sufficient conditions for the optimality of extremal controls that supplement the maximum principle. These conditions are represented in the form of inequalities and equalities for one-variable functions on a time interval. The optimization of a quadratic functional with the help of a matrix conjugate function is reduced to the bilinear case.

In this paper, we investigate interpolatory model order reduction for large-scale bilinear descriptor systems. Recently, it was shown in Wyatt et al. (2013) for linear descriptor systems that directly extending the standard rational interpolation conditions used in H 2 optimal model reduction to descriptor systems, in general, yields an unbounded error in the H 2 -norm. This is due to the possible mismatch of the polynomial part of the original and reduced-order systems. This conclusion holds for nonlinear systems as well. In this paper, we deal with bilinear descriptor systems and aim to pay attention to the polynomial part of the bilinear descriptor system along with interpolation. To this end, we have shown in Goyal et al. (2015) how to determine the polynomial part of each subsystem of the bilinear descriptor system explicitly, by assuming special structures of the system matrices. Considering the same structured bilinear descriptor systems, in this paper we first show how to achieve multipoint interpolation of the underlying Volterra series of bilinear descriptor systems while retaining the polynomial part of each subsystem of the bilinear system. Then, we extend the interpolation based first-order necessary conditions for H 2 optimality to bilinear descriptor systems and propose an iterative scheme to obtain an H 2 optimal reduced-order system. By means of various numerical examples, we demonstrate the efficiency of the proposed model order reduction technique and compare it with reduced bilinear systems obtained by using linear IRKA, the Loewner method for bilinear systems and POD-based approximations.

In this paper, we consider an optimal bilinear control problem for nonlinear Hartree equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first-order optimality system is rigorously derived. In particular, we prove Frechet-differentiability of the objective functional. Our results improve considerably some recent results.

In this paper, a stabilization problem of bilinear control systems is considered. Using the linear matrix inequality technique and quadratic Lyapunov functions, an approach is proposed to the construction of the so-called stabilizability ellipsoid such that the trajectories of the closed-loop system emanating from any point inside this ellipsoid asymptotically tend to the origin. The approach allows for an efficient construction of nonconvex approximations to stabilizability domains of bilinear systems. The results are extended to robust formulations of the problem, where the system matrix is subjected to structured uncertainty.

Bilinear systems are a special class of nonlinear systems. Some systems can be described by using bilinear models. This paper considers the parameter identification problems of bilinear stochastic systems. The difficulty of identification is that the model structure of the bilinear systems includes the products of the states and inputs. To this point, this paper gives the input---output representation of the bilinear systems through eliminating the state variables in the model and derives a least squares algorithm and a multi-innovation stochastic gradient algorithm for identifying the parameters of bilinear systems based on the least squares principle and the multi-innovation identification theory. The simulation results indicate that the proposed algorithms are effective for identifying bilinear systems.

Linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations are analyzed. Necessary and sufficient conditions are presented to apply the linear superposition principles. Applications of an algorithm using weights are made together with a few concrete examples.

In this paper, we consider an optimal bilinear control problem for the Gross-Pitaevskii equations with Coulombian potentials. We show the well-posedness of the problem and the existence of an optimal control. In addition, the first order optimality system is rigorously derived. In particular, we prove the Frechet-differentiability of the unconstrained functional. We extend the study of Hintermuller et al. (2013) [15] to more general power nonlinearities and unbounded potentials. (C) 2015 Elsevier Inc. All rights reserved.

We propose a vector optimization approach to linear Cournot oligopolistic market equilibrium models where the strategy sets depend on each other. We use scalarization technique to find a Pareto efficient solution to the model by using a jointly constrained bilinear programming formulation. We then propose a decomposition branch-and-bound algorithm for globally solving the resulting bilinear problem. The subdivision takes place in one-dimensional intervals that enables solving the problem with relatively large sizes. Numerical experiments and results on randomly generated data show the efficiency of the proposed algorithm.

We consider the problem of practical stabilization of bilinear third-order dynamical systems by stationary state feedback. For bilinear systems in canonical form, we suggest to generalize the method of feedback linearization on the basis of a feedback of variable structure. This generalization is used for the derivation of sufficient conditions for the practical stabilization of the zero equilibrium of bilinear third-order systems in canonical form.

In this paper, the binary Bell polynomials are employed to find the bilinear form, bilinear Backlund transformation and Lax pair for the (3+1)-dimensional BKP equation. Based on Hirota’s bilinear form and three-wave method, multi-soliton solutions are presented. Furthermore, a new bilinear Backlund transformation is constructed via applying a gauge transformation to the Backlund transformation in bilinear form.

With symbolic computation, Bell-polynomial scheme and bilinear method are applied to a two-dimensional Korteweg–de Vries (KdV) model, which is firstly proposed with Lax pair generating technique. Bell-polynomial expression with one auxiliary independent variable is derived and transformed into bilinear form. According to the coupled two-field conditions between the primary and replica fields, Bell-polynomial-typed Backlund transformations (BTs) are constructed and converted into the bilinear ones. Finally, soliton solutions of the two-dimensional KdV model are obtained (via solving the bilinear representation and BT, respectively) and compared. Such associated integrable properties as bilinear representation, BT (especially auxiliary-independent-variable-involved Bell-polynomial-typed ones constructed in this paper) and soliton solutions (especially the multi-soliton ones) may be useful for further study on other two-dimensional KdV and KdV-typed models.

We investigate behavior of solutions of the following systems of rational difference equations: xn+1=yn-3k-1/(±1±yn-(3k-1)xn-(2k-1)yn-(k-1)),yn+1=xn-3k-1/(±1±xn-3k-1yn-2k-1xn-k-1), where k is a positive integer and the initial conditions are real numbers. We show that every solution is periodic with 6k period, considerably improving the results in the literature.

This paper formulates optimal bilinear observers for bilinear state-space models. Observers in bilinear form, as opposed to other nonlinear forms, are required to develop an extension of observer/Kalman filter identification for simultaneous identification of a bilinear state-space model and an associated bilinear observer from noisy input–output measurements. The paper establishes the relationship between the bilinear observer gains and the interaction matrices which are used to convert the original bilinear state-space model to a form that simplifies the identification of such a model. Techniques to find the interaction matrices are developed. In the absence of noises, these matrices produce the gains of the fastest converging observer. In the presence of noises, they minimise the state estimation error in the same manner as a standard steady-state Kalman filter. Numerical examples illustrate both the theoretical and computational aspects of the proposed algorithms.

本文主要是对两个方程组的实数解的个数的判定，可知方程组(1.a)在实数范围内没有实数解，方程组(1.b)在实数范围内有四个解。 In this paper, by determining the number of real solutions to the two equations, we can know that there is no real solution for the equations (1.a) in real numbers, and there are four solutions for the equation (1.b).

We represent a general bilinear Calderón--Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so called sparse $T1$ theorem for bilinear singular integrals., Juan de la Cierva - Formaci\'on 2015 FJCI-2015-24547, Academy of Finland grants 294840 and 306901, National Science Foundation Grant No. DMS-1440140.

Consider the following system of difference equations:xn+1(i)=xn-m+1(i)/Ai∏j=0m-1xn-j(i+j+1)+αi,xn+1(i+m)=xn+1(i),x1-l(i+l)=ai,l,Ai+m=Ai,αi+m=αi,i,l=1,2,…,m;n=0,1,2,…,wheremis a positive integer,Ai,αi,i=1,2,…,m, and the initial conditionsai,l,i,l=1,2,…,m,are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.

It is shown that the approximate bilinear complexity of multiplying matrices of the order 2 × 2 by a matrix of the order 2 × 6 does not exceed 19. An approximate bilinear algorithm of complexity 19 is presented for this task.

It is well known that estimating bilinear models is quite challenging. Many different ideas have been proposed to solve this problem. However, there is not a simple way to do inference even for its simple cases. This article proposes a generalized autoregressive conditional heteroskedasticity-type maximum likelihood estimator for estimating the unknown parameters for a special bilinear model. It is shown that the proposed estimator is consistent and asymptotically normal under only finite fourth moment of errors.

In this technical note, a new method for model reduction of bilinear systems is presented. The method is developed in particular for many applications in which one is interested to approximate a system in a given frequency-interval. To this end, new generalized frequency-interval gramians are introduced for bilinear systems. It is shown that these gramians are the solutions to the so-called frequency-interval generalized Lyapunov equations. The conditions for these equations to be solvable are derived and an algorithm is proposed to solve such equations iteratively. The method is further illustrated with the help of an illustrative example. The numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians.

Bilinear derivative method is widely used in calculating multi-soliton solutions of some nonlinear evolution equation. The paper proves some frequently used properties of bilinear derivative from the perspective of the definition of bilinear derivative, hoping to be useful for learning and teaching in nonlinear science.

Near-controllability is defined for those systems that are uncontrollable but have a large controllable region. It is a property of nonlinear control systems introduced recently, and it has been demonstrated on two classes of discrete-time bilinear systems. This paper studies near-controllability of discrete-time upper-triangular bilinear systems, which are uncontrollable and are more general than the two classes mentioned. A necessary and sufficient condition for the systems in dimension two to be nearly controllable is presented, which covers the existing results. For the systems with high dimensions, necessary conditions and sufficient conditions of near-controllability are provided, which generalize the existing results. In particular, the obtained near-controllability results are applied to controllability of discrete-time bilinear systems. An example also is given to demonstrate the effectiveness, which shows that the controllability problems of discrete-time bilinear systems can be solved by near-controllability.

Commutators of bilinear pseudodifferential operators with symbols in the Hormander class $BS_{1, 0}^{1}$ and multiplication by Lipschitz functions are shown to be bilinear Calderon-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.

A unified identification method is proposed for system realization of a deterministic continuous-time/discrete-time bilinear models from input and output measurement data. A generalized Hankel matrix is formed with the output measurements obtained by applying a set of repeated input sequences to a bilinear system. A computational procedure is developed to extract a time varying discrete-time state-space model from the generalized Hankel matrix. The bilinear system models are realized by transforming the identified time varying discrete-time model to the bilinear models. Numerical simulations are given to show the effectiveness of the proposed identification method.

In this paper, the (2+1)-dimensional Sawada–Kotera model is studied with the Hirota bilinear method, gauge transformation and symbolic computation. Based on an alternative bilinear representation of the model, a bilinear Backlund transformation (BT) with three arbitrary constants is derived. Via applying a gauge transformation to this BT and choosing suitable constant parameters, three other sets of bilinear BTs are constructed, among which, the last set is treated as a new bilinear BT and denoted as BTIV hereby. Finally, by performing the perturbation technique on the new bilinear BT, namely BTIV, multisoliton solutions are iteratively achieved, and as an example, the one-, two- and three-soliton solutions are explicitly given. Note that formulas of the soliton solutions obtained hereby through solving the BTIV are different from the previous ones in other literature.

We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C^4 x C^4 -> C^4. This space of bilinear maps includes the matrix multiplication operator M_2 for two by two matrices. We show these newly obtained polynomials do not vanish on the matrix multiplication operator M_2, which gives a new proof that the border rank of the multiplication of 2 x 2 matrices is seven. Other examples are considered along with an explanation of how to implement the methods., 13 pages

Relationship between bilinear algorithms over fields of different characteristics and over different rings is investigated.

Summary form only given. This paper generalizes the recently introduced bilinear formulation of the WLS state estimation problem to those cases in which equality constraints must be explicitly considered. This leads to augmented equation systems in which both state variables and Lagrange multipliers get updated throughout the three stages arising in the bilinear approach (two linear filters with a nonlinear transformation in between). The proposed formulation prevents the ill-conditioning typically arising when exact-injection constraints are handled as virtual measurements with huge weights, while the excellent convergence speed of the bilinear scheme, which for practical purposes reaches the optimal solution in a single iteration, is fully preserved.

A novel method is developed incorporating a special input sequence for a general class of observable and reachable bilinear system identification. An order reduction scheme is introduced to determine the system matrices. This method makes a significant improvement to its predecessors by removing the observability requirement for the linear part of the bilinear systems. Numerical examples are given to demonstrate the method developed in this paper.

We propose a notion of orbit for bilinear mappings on Banach spaces. This then leads us to the concept of bihypercyclic bilinear mappings, that is, of bilinear mappings with a dense orbit. We provide some examples of such mappings, and we obtain some basic properties of bihypercyclicity. In our main result we show that every separable (real or complex) Banach space supports a bihypercyclic bilinear mapping. We also formulate some open problems.

International audience; This paper presents an energy estimate in terms of the total variation of the control for bilinear infinite dimensional quantum systems with unbounded potentials. These estimates allow a rigorous construction of propagators associated with controls of bounded variation. Moreover, upper bounds of the error made when replacing the infinite dimensional system by its finite dimensional Galerkin approximations is presented.

Binary Bell polynomials are applied to construct two kinds of bilinear derivative equations of a $(3+1)$ -dimensional nonlinear equation. Based on one of the bilinear forms, we derive a Backlund transformation, the corresponding Lax pair, infinite conservation laws, and explicit solutions with an arbitrary function in y. In the meantime, from the other bilinear form, we get another bilinear Backlund transformation and exact solutions by utilizing the exchange formulas for Hirota’s bilinear operators.

We provide some remarks concerning a bilinear square function formed by products of Littlewood–Paley operators over arbitrary intervals. For 1 2∕3 and unbounded when p < 2∕3.

Optimal control of bilinear systems has been a well-studied subject in the areas of mathematical and computational optimal control. However, effective methods for solving emerging optimal control problems involving an ensemble of deterministic or stochastic bilinear systems are underdeveloped. These burgeoning problems arise in diverse applications from quantum control and molecular imaging to neuroscience. In this work, we develop an iterative method to find optimal controls for an inhomogeneous bilinear ensemble system with free-endpoint conditions. The central idea is to represent the bilinear ensemble system at each iteration as a time-varying linear ensemble system, and then solve it in an iterative manner. We analyze convergence of the iterative procedure and discuss optimality of the convergent solutions. The method is directly applicable to solve the same class of optimal control problems involving a stochastic bilinear ensemble system driven by independent additive noise processes. We demonstrate the robustness and applicability of the developed iterative method through practical control designs in neuroscience and quantum control., Comment: 12 pages, 5 figures

We consider the stabilization problem for the zero equilibrium of bilinear and affine systems in canonical form. We obtain necessary and sufficient conditions for the stabilizability of second-order bilinear and affine systems in canonical form and generalize these conditions to the case of simultaneous stabilization of a family of bilinear systems and to nth-order bilinear systems.

In this paper the Wiener–Hammerstein Benchmark is identified as a bilinear discrete system. The bilinear approximation relies on both facts that the Wiener–Hammerstein system can be described by a Volterra series which can be approximated by bilinear systems. The identification is performed with an iterative bilinear subspace identification algorithm previously proposed by the authors. In order to increase accuracy, polynomial static nonlinearities were added to the bilinear model input. These Hammerstein type bilinear models are then identified using the same iterative subspace identification algorithm.

The dual purpose of this article is to establish bilinear Poincare-type estimates associated with an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato–Morrey spaces under Sobolev scaling.

New exact Grammian determinant solutions to two (3+1)-dimensional nonlinear evolution equations are derived. Extended set of sufficient conditions consisting of linear partial differential equations with variable-coefficients is presented. Moreover, a systematic analysis of linear partial differential equations is used for solving the representative linear systems. The bilinear Backlund transformations are also constructed for the equations. Also, as an application of the bilinear Backlund transforms, a new class of wave and rational solutions to the equations are explicitly computed.

SUMMARY In this paper, we consider the problem of Nash equilibrium solutions of two players tracking game for bilinear systems. A successive approximation approach is modified to design optimal controllers for bilinear systems. More specifically, a sequence of extended Sylvester differential equations are solved by this modified approach. A simulation example is given to demonstrate the validity of this approach. Copyright © 2012 John Wiley & Sons, Ltd.

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.

We explore when Hirota bilinear equations possess linear subspaces of solutions. First, we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to Hirota bilinear equations. Second, we show that multivariate polynomials whose zeros form a vector space can generate the desired Hirota bilinear equations with given linear subspaces of solutions, and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero. Third, applying an algorithm using weights, we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired Hirota bilinear equations.

This article studies the realisability property of discrete-time bilinear and quadratic input–output (i/o) equations in the classical state-space form. Constraints on the parameters of the i/o model are suggested that lead to realisable models. Using new formulae for computing basis vectors of certain vector spaces of differential one-forms, we present in this article the complete list of the third- and fourth-order realisable i/o bilinear models, and a new realisable subclass of an arbitrary order is suggested. Moreover, we provide the sufficient conditions of the second- and third-order realisable i/o quadratic models, respectively. All the developed theory and algorithms are illustrated by means of several examples.

In this paper, necessary and sufficient conditions for the controllability of a class of discrete-time bilinear systems are proposed, which extend the existing results and show that the controllability counterexample in [1] that is derived by Euler discretization of an uncontrollable bilinear system, is a special case of the necessary and sufficient conditions.